1. Field of the Invention
The present invention relates generally to a method of predicting the service life of filled polymeric materials.
2. Description of the Related Art
There are numerous situations in which it is important to be able to predict the fatigue failure of a filled polymeric material. Examples of filled polymeric materials in which fatigue failure is critical include airplane parts, rubber tires and solid propellant rocket motors. Since in many cases the polymer will have a long service life, perhaps on the order of years, it is desirable and often critical to be able to estimate the service life using mathematical models.
For example, in the case of solid propellant rocket motors, grain structural integrity can be the factor limiting the usable service life. If structural failure occurs, it is almost certain that ballistic performance will be substantially altered, possibly to the point of catastrophic motor failure. It is therefore desirable to be able to estimate what the chances of grain failure are as the motor is handled and stored prior to use.
In the case of solid propellant rocket motors, there are many modes by which the solid propellant within the rocket motor can receive a mechanical stress load. One of the most common is stress arising from thermal contraction. This particularly significant where the propellant is bonded to a steel pressure vessel. The propellant is typically cured at elevated temperatures to chemically accelerate the curing process. Since the coefficient of thermal expansion of the propellant is typically an order of magnitude greater than that of the steel vessel, and the Young's modulus of steel is roughly five orders of magnitude higher than that of the propellant, the propellant cannot contract fully upon cooling. This yields a rocket motor which is stress-free at the elevated temperature, but under continual stress once the rocket motor cools.
The modeling of stress in this situation is complicated as the propellant, in time, responds to this environment by undergoing changes at the molecular level that tend to relieve some of this stress. These changes generally consist of viscous flow of the polymeric binder and changes in crosslinking. These changes cause the propellant to take a permanent set, much as a garden hose which has been coiled in storage. This process is commonly referred to as a shift in the stress-free temperature.
While stressed, the binder microstructure also begins to tear. This process has been observed to increase linearly with time up to the point of macroscopic fracture when the applied stress is constant. The degree of damage, or damage fraction, done to the binder is therefore directly related to the total time the stress was applied and the time required to produce macroscopic failure at that same stress, specifically to the ratio of the former to the latter. At different levels of stress the amount of damage produced in a given amount of time varies considerably, being proportional to the stress raised to a large power, usually in the range of 6 to 12. If the propellant has been loaded to a number of different stresses, that is, has a complex service life history, the cumulative damage is simply the sum of each of the individual constant stress components. When they sum to unity, macroscopic failure is imminent.
The binder may also continue to undergo chemical changes long after the motor has been removed from its curing oven. These may include continued crosslinking chemical reactions with trace amounts of curative or reactions induced by exposure to the ambient environment, for example binder oxidation. Migration of mobile chemical species may produce non-homogeneous areas with the grain. These will frequently manifest themselves as changes in the propellant's mechanical properties which will in turn modify the level of stress, by changes in Young's modulus, or the strength, by changes in the maximum stress that can be attained.
To account for the inevitable cyclic nature of the loading in a complex sequence the linear cumulative damage model is often employed. As the name implies, there is a finite amount of damage sustained by the material during each segment of its load history. The damage contributions are numerically added, using a running total. Damage is conveniently expressed as a ratio, defined as the time dwelt under a constant load divided by the time required to produce failure at the same load level. When the sum of damage for a many-load sequence approaches unity, failure is imminent.
The level of stress within the structure must be known in order to estimate damage. This is usually determined by performing a finite element analysis. For complicated structures the corresponding finite element model may be quite large, requiring a significant amount of computation to exercise. If the load sequence is long and varied it may be necessary to make many runs of the finite element model to compute the corresponding stress sequence. For moderate to large finite element models, the time needed is often so large that it is not practical to perform the calculation. Rather, engineering judgments and approximations are sometimes made, so that many of the loads suspected of causing little or no damage are ignored. This is at best an imprecise process.
There is another complicating factor that is specifically associated with polymeric materials: the large statistical variability of the mechanical properties. Furthermore, when estimating the performance of a large population of structural members, the specific environment for each one may not be the same, but rather lies within some statistical distribution. These factors place a large uncertainty in the level stress applied within the structure throughout its service life. Because the amount of damage changes exponentially as the stress changes, this uncertainty in the stress magnitude is magnified in the results.
While the older models can accurately evaluate the amount of damage that occurs over a specific set of material properties and environmental conditions there has not been any provision to gauge the impact of their statistical variability. Although this could be done by simply repeating the analysis a number of times, using a statistical sample input for each and noting the incidence of failure and success, this has been not practical owing to the large amount of computational effort required. For structures with a relatively low failure rate, which is the usual case, the finite element model would need to be exercised thousands or even tens of thousands of times, that is, once for each load level within each load history, to estimate reliability. For a finite element model of any appreciable size this is not practical and practice is not done. Rapid and easy calculation of the fatigue life would be generally desirable, and would be particularly useful in a design process where the designer may have to produce numerous design iterations.
Methods to predict fatigue failure are seen in the following examples of the contemporary art. U.S. Pat. No. 5,531,123, to Henkel, entitled Remote and Powerless Miniature Fatigue Monitor and Method, describes a passive monitor for measurement of fatigue and a method for fatigue testing. The method includes locating a region of high stress, monitoring the fatigue in the direction of principal stress using a passive fatigue monitor during cycle loads, and using Miner's rule for cumulative damage to estimate fatigue life. This method, however, requires the identification of a high stress region and the experimental cycling of the material under loads to obtain fatigue data. This is an involved process which requires assumptions about the regions of high stress, and is in particular not practical for rapid estimation of fatigue life in a design process.
U.S. Pat. No. 5,736,645, to Chin-Chan et al., entitled Method of Predicting Crack Initiation Based Fatigue Life, describes a method involving plotting the finite element stress states of a stress history for a given component at a predetermined critical location in the stress space and determining the least square fit ellipsoid of the stress states. This method attempts to overcome the problems of uniaxial fatigue analysis by performing a multiaxial fatigue analysis. We have found that this method does not address many of the problems associated with filled polymeric materials, however.
"Problem of the Month, July 1997--Monte Carlo Reliability Model Starting With WinSMITH Weibull Data", originating with Barringer and Associates, Inc., describes a method for estimating the fatigue life of a pressure vessel (coke drum). The method involves obtaining stress data, building (modeling) the stress distribution, simulating stress using a Monte Carlo approach and the use of Miner's rule for fatigue. This example illustrates the use of the Miner's rule assumption that when the sum of damage for a many-load sequence approaches unity, failure is imminent. In this example, though, a model having actual experimental stresses and the S-N curve was available, and a complete finite element analysis was not necessary for each cycle. We have noticed this example therefore does not address the problems encountered in using a Monte Carlo approach with filled polymeric materials.
We have discovered, then, that what is needed is an improved, more rapid method of estimating the probability of fatigue failure in filled polymeric materials.